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International Journal of Engineering Sciences Paradigms and Researches, Vol. 02, Issue 01, December 2012ISSN (Online): 2319-6564www.ijesonline.com

Analysis of Brain Cancer and Nervous Cancer Population with Age and Brain Cancer Type

In North America

Akash K Singh, PhDIBM Corporation, Sacramento, USA

[email protected]

AbstractThis paper discuss about statistical representation of Brain Cancer in America. Brain cancer morbidity is high and treatment plans like chemotherapy, surgical resection of Tumor, Hyperthermia and Radio Surgery is key elements for the treatment of patients suffering from Brain Cancer. Who and Disease control prevention dataset is used to perform analysis. Incidence Rate, Death Rate, Incidence Count and Death count in male and female are rising; Classification of Data is based on Brain Tumor and other Nervous. Brain Tumor is a leading cause of death and once its diagnosed base on the stage of cancer life expectancy is about 5 Years or so. Incidence rate of Brain Cancer in age group and gender difference is analyzed based on States. Spatially Analytic data is used for the geo-visualization of Cancer. Sources of data are from Cancer registries, World Health Organization, Health Information Database and remote sensing data.Keywords: Brain Cancer, Spatial Analysis, Autocorrelation, Fuzzy Logic.

I. INTRODUCTIONThis research is focus on giving tools and techniques to the field

of epidemiology to study and provide treatment to Brain Cancer patient. This would help to control the disease and create the disease model and act on the trends of Brain Cancer. Large and highly complex data structure are analysed on grid computing environment. Purpose of this research is to provide the growth of Brain Cancer in America and find out the similarity and differences in the regions of Brain Cancer depending upon spatial information. Geo Spatial information helps in predicting the spread of disease. Mathematical model helps in analysing the Brain cancer Characteristics. Cancer Etiology is also represented in spatial form and pattern on Treatment [1]. Spatial data refer to data with locational attributes. Most commonly, locations are given in Cartesian coordinates referenced to the earth's surface. These coordinates may describe points, lines, areas or volumes. This need not be the only spatial framework; "relative spaces" may be defined in which distance is defined in terms of some other attribute, such as socio-demographic similarly or connectedness along transportation networks [2][3]. There are over 600,000 people in the US living with a primary brain tumor and over 28,000 of these cases are among children under the age of 20.1

Metastatic brain tumors (cancer that spreads from other parts of the body to the brain) occur at some point in 20 to 40% of persons with cancer and are the most common type of brain tumor.

Over 7% of all reported primary brain tumors in the United States are among children under the age of 20.

Each year approximately 210,000 people in the United States are diagnosed with a primary or metastatic brain tumor. That's over 575 people a day:

An estimated 62,930 of these cases are primary malignant and non-malignant tumors.

The remaining cases are brain metastases (cancer that spreads from other parts of the body to the brain).

Among children under age 20, brain tumors are:the most common form of solid tumor the second leading cause of cancer-related deaths, following leukemiathe second leading cause of cancer-related deaths among females

Among adults, brain tumors are:

the second leading cause of cancer-related deaths among males up to age 39the fifth leading cause of cancer-related deaths among women ages 20-39

There are over 120 different types of brain tumors, making effective treatment very complicated. Because brain tumors are located at the control center for thought, emotion and movement, their effects on an individual's physical and cognitive abilities can be devastating. At present, brain tumors are treated by surgery, radiation therapy, and chemotherapy, used either individually or in combination. No two brain tumors are alike. Prognosis, or expected outcome, is dependent on several factors including the type of tumor, location, response to treatment, an individual's age, and overall health status.

IJESPRwww.ijesonline.com

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An estimated 35% of adults living with a primary malignant brain or CNS tumor will live five years or longer.

Brain tumors in children are different from those in adults and are often treated differently. Although over 72% percent of children with brain tumors will survive, they are often left with long-term side effects [4].

II. METHODOLOGYStudy of spatial autocorrelation analysis supports the hypotheses

to predict the geo location and volume of epidemiological insights. Information pertained from first order autocorrelation (Brain Cancer & Nervous) gives the pattern of mortality in spatial space.Applied spatial autocorrelation to define correlation of a cancer dataset in variable array with itself through Fuzzy Topological space. Measured the characteristics at one state example California are similar or dissimilar to nearby states example Nevada. Measure the most probable occurrence of event at one location with nearby inter-connected locations.Applied the measurement using Joint Count Statistics, Moran’s I , Geary’s ratio, General G, Local Index of Spatial Autocorrelation and Global Index of Spatial Autocorrelation.Spatial Autocorrelation produced positive results with similar values Fuzzy Cluster on the map and Negative dissimilar values Fuzzy Cluster on the map. Fuzzy connectedness technique is used to measure brain tumor volume; this is also applied on brain lesion volume estimation. Multiple Fuzzy spaces are defined to layout the computational framework. Fuzzy compactness and connectedness are distinct absolute property that is used for fuzzy topology. Absolute topology is where all subspaces Z Y X of a space X, Z fulfills P (property) a subspace of Y iff Z fulfills P as a subspace of X. We consider the following anycast field equations defined over an open bounded piece of network and /or feature space

. They describe the dynamics of the mean anycast of each of node populations.

We give an interpretation of the various parameters and functions that appear in (1), is finite piece of nodes and/or feature space

and is represented as an open bounded set of . The vector

and represent points in . The function is the normalized sigmoid function:

It describes the relation between the input rate of population as a function of the packets potential, for example,

We note the dimensional

vector The function represent the

initial conditions, see below. We note the dimensional

vector The function represent

external factors from other network areas. We note the

dimensional vector The matrix of functions

represents the connectivity between populations

and see below. The real values determine the threshold of activity for each population, that is, the value of the nodes potential corresponding to 50% of the maximal activity.

The real positive values determine the slopes of the sigmoids at the origin. Finally the real positive values

determine the speed at which each anycast node potential decreases exponentially toward its real value. We also

introduce the function defined by

and the diagonal

matrix Is the intrinsic dynamics of the population given by the linear response of data transfer.

is replaced by to use the alpha function

response. We use for simplicity although our analysis applies to more general intrinsic dynamics. For the sake, of generality, the propagation delays are not assumed to be identical

for all populations, hence they are described by a matrix

whose element is the propagation delay between

population at and population at The reason for this assumption is that it is still unclear from anycast if propagation delays are independent of the populations. We assume for technical

reasons that is continuous, that is Moreover packet data indicate that is not a symmetric function

i.e., thus no assumption is made about this symmetry unless otherwise stated. In order to compute the

righthand side of (1), we need to know the node potential factor

on interval The value of is obtained by considering the maximal delay:


30


Hence we choose

III. MATHEMATICAL FRAMEWORKA convenient functional setting for the non-delayed packet field

equations is to use the space which is a Hilbert space endowed with the usual inner product:

To give a meaning to (1), we defined the history space

with which is the Banach phase space associated with equation (3). Using the

notation we write (1) as

Where

Is the linear continuous operator satisfying Notice that most of the papers on this subject assume infinite,

hence requiring

Proposition 1.0 If the following assumptions are satisfied.

1.

2. The external current

3.

Then for any there exists a unique solution

to (3)

Notice that this result gives existence on finite-time explosion is impossible for this delayed differential equation. Nevertheless, a particular solution could grow indefinitely, we now prove that this cannot happen.

Boundedness of SolutiONS

A valid model of neural networks should only feature bounded packet node potentials.

Theorem 1.0 All the trajectories are ultimately bounded by the

same constant if

Proof :Let us defined as

We note

Thus, if

Let us show that the open route of of center 0 and radius

is stable under the dynamics of equation. We know that

is defined for all and that on the

boundary of . We consider three cases for the initial condition

If and set

Suppose that then is defined and belongs to

the closure of because is closed, in effect to we

also have because

Thus we deduce that for and small enough,

which contradicts the definition of T. Thus

and is stable. Because f<0 on

implies that . Finally we consider the case

. Suppose that then

thus is monotonically

decreasing and reaches the value of R in finite time when

reaches This contradicts our assumption. Thus


31


Proposition 1.1 : Let and be measured simple functions on

for define

Then is a measure on .

Proof : If and if are disjoint members of whose

union is the countable additivity of shows that

Also, so that is not identically .

Next, let be as before, let be the distinct values of

t,and let If the

and Thus (2) holds

with in place of . Since is the disjoint union of the sets

the first half of our proposition implies that (2) holds.

Theorem 1.1: If is a compact set in the plane whose

complement is connected, if is a continuous complex function

on which is holomorphic in the interior of , and if then

there exists a polynomial such that for all

. If the interior of is empty, then part of the hypothesis is

vacuously satisfied, and the conclusion holds for every . Note that need to be connected.

Proof: By Tietze’s theorem, can be extended to a continuous function in the plane, with compact support. We fix one such

extension and denote it again by . For any let be

the supremum of the numbers Where and

are subject to the condition . Since is uniformly

continous, we have From now on,

will be fixed. We shall prove that there is a polynomial such that

By (1), this proves the theorem. Our first objective is the

construction of a function such that for all

And

Where is the set of all points in the support of whose

distance from the complement of does not . (Thus contains no point which is “far within” .) We construct as the

convolution of with a smoothing function A. Put if

put

And define

For all complex . It is clear that . We claim that

The constants are so adjusted in (6) that (8) holds. (Compute the integral in polar coordinates), (9) holds simply because has

compact support. To compute (10), express in polar

coordinates, and note that

Now define


32


Since and have compact support, so does . Since

And if (3) follows from (8). The difference quotients of converge boundedly to the corresponding partial

derivatives, since . Hence the last expression in (11) may be differentiated under the integral sign, and we obtain

The last equality depends on (9). Now (10) and (13) give (4). If we

write (13) with and in place of we see that has

continuous partial derivatives, if we can show that in

where is the set of all whose distance from the

complement of exceeds We shall do this by showing that

Note that in , since is holomorphic there. Now if

then is in the interior of for all with The mean value property for harmonic functions therefore gives, by the first equation in (11),

For all , we have now proved (3), (4), and (5) The definition of shows that is compact and that can be

covered by finitely many open discs of radius

whose centers are not in Since is connected, the

center of each can be joined to by a polygonal path in

. It follows that each contains a compact connected set

of diameter at least so that is connected and so

that with . There are functions

and constants so that the inequalities.

Hold for and if

Let be the complement of Then is an open

set which contains Put and

for

Define And

Since,

(18) shows that is a finite linear combination of the functions

and . Hence By (20), (4), and (5) we have

Observe that the inequalities (16) and (17) are valid with in

place of if and Now fix , put

and estimate the integrand in (22) by (16) if

by (17) if The integral in (22) is then seen to be less than the sum of

And

Hence (22) yields


33


Since and is connected, Runge’s theorem shows that can be uniformly approximated on by polynomials. Hence (3) and (25) show that (2) can be satisfied. This completes the proof.

Lemma 1.0 : Suppose the space of all continuously differentiable functions in the plane, with compact support. Put

Then the following “Cauchy formula” holds:

Proof: This may be deduced from Green’s theorem. However, here is a simple direct proof:

Put real

If the chain rule gives

The right side of (2) is therefore equal to the limit, as of

For each is periodic in with period . The integral

of is therefore 0, and (4) becomes

As uniformly. This gives (2)

If and , then

, and so satisfies the condition . Conversely,

and so if satisfies , then the subspace generated by the

monomials , is an ideal. The proposition gives a

classification of the monomial ideals in : they are in

one to one correspondence with the subsets of satisfying

. For example, the monomial ideals in are exactly the

ideals , and the zero ideal (corresponding to the

empty set ). We write for the ideal

corresponding to (subspace generated by the ).

LEMMA 1.1. Let be a subset of . The the ideal

generated by is the monomial ideal corresponding to

Thus, a monomial is in if and only if it is divisible by one of the

PROOF. Clearly satisfies , and .

Conversely, if , then for some , and

. The last statement follows from the fact

that . Let satisfy . From the geometry of , it is clear that there is a finite set of elements

of such that

(The are

the corners of ) Moreover, is generated by

the monomials .

DEFINITION 1.0. For a nonzero ideal in ,

we let be the ideal generated by

LEMMA 1.2 Let be a nonzero ideal in ; then

is a monomial ideal, and it equals

for some .

PROOF. Since can also be described as the ideal generated by the leading monomials (rather than the leading terms) of elements of .


34


THEOREM 1.2. Every ideal in is finitely

generated; more precisely, where are

any elements of whose leading terms generate

PROOF. Let . On applying the division algorithm, we

find , where either or no monomial occurring in it is divisible by

any . But , and therefore

, implies that every

monomial occurring in is divisible by one in . Thus

, and .

DEFINITION 1.1. A finite subset of an ideal

is a standard ( bases for if

. In other words, S is a standard basis if the leading term of every element of is divisible by at

least one of the leading terms of the .

THEOREM 1.3 The ring is Noetherian i.e., every ideal is finitely generated.

PROOF. For is a principal ideal domain, which means that every ideal is generated by single element. We shall prove the theorem by induction on . Note that the obvious map

is an isomorphism – this

simply says that every polynomial in variables

can be expressed uniquely as a polynomial in with coefficients

in :

Thus the next lemma will complete the proof

LEMMA 1.3. If is Noetherian, then so also is PROOF. For a polynomial

is called the degree of , and is its leading coefficient. We call 0 the leading coefficient of the polynomial 0. Let be an

ideal in . The leading coefficients of the polynomials in

form an ideal in , and since is Noetherian, will be

finitely generated. Let be elements of whose leading

coefficients generate , and let be the maximum degree of .

Now let and suppose has degree , say,

Then , and so we can write

Now

has degree . By continuing in this way, we find that

With a polynomial of degree

. For each , let be the subset of consisting of 0

and the leading coefficients of all polynomials in of degree

it is again an ideal in . Let be polynomials of

degree whose leading coefficients generate . Then the same

argument as above shows that any polynomial in of degree

can be written With

of degree . On applying this remark repeatedly we

find that Hence

and so the polynomials generate

One of the great successes of category theory in computer science has been the development of a “unified theory” of the constructions underlying denotational semantics. In the untyped -calculus, any term may appear in the function position of an application. This means that a model D of the -calculus must have the

property that given a term whose interpretation is Also, the interpretation of a functional abstraction like . is most

conveniently defined as a function from , which must then

be regarded as an element of D. Let be the function that picks out elements of D to represent elements of

and be the function that maps

elements of D to functions of D. Since is intended to


35


represent the function as an element of D, it makes sense to

require that that is, Furthermore, we often want to view every element of D as representing some function from D to D and require that elements representing the same function be equal – that is

The latter condition is called extensionality. These conditions

together imply that are inverses--- that is, D is isomorphic to the space of functions from D to D that can be the

interpretations of functional abstractions: .Let us suppose we are working with the untyped , we need

a solution ot the equation where A is some predetermined domain containing interpretations for elements of C. Each element of D corresponds to either an element of A or an

element of with a tag. This equation can be solved by

finding least fixed points of the function from domains to domains --- that is, finding domains X such that

and such that for any domain Y also satisfying this equation, there is an embedding of X to Y --- a pair of maps

Such that

Where means that in some ordering representing their information content. The key shift of perspective from the domain-theoretic to the more general category-theoretic approach lies in considering F not as a function on domains, but as a functor on a category of domains. Instead of a least fixed point of the function, F.

Definition 1.3: Let K be a category and as a functor. A fixed point of F is a pair (A,a), where A is a K-object and

is an isomorphism. A prefixed point of F is a pair (A,a), where A is a K-object and a is any arrow from F(A) to ADefinition 1.4 : An in a category K is a diagram of the following form:

Recall that a cocone of an is a K-object X and a

collection of K –arrows such that

for all . We sometimes write as

a reminder of the arrangement of components Similarly, a

colimit is a cocone with the property that if

is also a cocone then there exists a unique mediating

arrow such that for all . Colimits of are sometimes referred to as .

Dually, an in K is a diagram of the following form:

A cone of an

is a K-object X and a collection of K-arrows

such that for all . An

-limit of an is a cone with the

property that if is also a cone, then there exists a

unique mediating arrow such that for all

. We write (or just ) for the distinguish initial object of K, when it has one, and for the unique arrow from to each K-object A. It is also convenient to write

to denote all of except and .

By analogy, is . For the images of and under F we write

and

We write for the i-fold iterated composition of F – that is,

,etc. With these definitions we can state that every monitonic function on a complete lattice has a least fixed point:

Lemma 1.4. Let K be a category with initial object and let

be a functor. Define the by


36


If both and are colimits,

then (D,d) is an intial F-algebra, where is the

mediating arrow from to the cocone

Theorem 1.4 Let a DAG G given in which each node is a random variable, and let a discrete conditional probability distribution of each node given values of its parents in G be specified. Then the product of these conditional distributions yields a joint probability distribution P of the variables, and (G,P) satisfies the Markov condition.

Proof. Order the nodes according to an ancestral ordering. Let

be the resultant ordering. Next define.

Where is the set of parents of of in G and is the specified conditional probability distribution. First we show this does indeed yield a joint probability distribution. Clearly,

for all values of the variables. Therefore, to show we have a joint distribution, as the variables range through all their possible values, is equal to one. To that end, Specified conditional distributions are the conditional distributions they notationally represent in the joint distribution. Finally, we show the Markov condition is satisfied. To do this, we need show for

that

whenever

Where is the set of nondescendents of of in G. Since

, we need only show .

First for a given , order the nodes so that all and only

nondescendents of precede in the ordering. Note that this

ordering depends on , whereas the ordering in the first part of the proof does not. Clearly then

follows

We define the cyclotomic field to be the field

Where is the cyclotomic

polynomial. has degree over

since has degree . The roots of are just the

primitive roots of unity, so the complex embeddings of

are simply the maps

being our fixed choice of primitive root of unity. Note that

for every it follows that for all

relatively prime to . In particular, the images of the

coincide, so is Galois over . This means that

we can write for without much fear of ambiguity; we will do so from now on, the identification being

One advantage of this is that one can easily talk about cyclotomic fields being extensions of one another,or intersections or compositums; all of these things take place considering them as

subfield of We now investigate some basic properties of cyclotomic fields. The first issue is whether or not they are all distinct; to determine this, we need to know which roots of unity lie

in .Note, for example, that if is odd, then is a

root of unity. We will show that this is the only way in which

one can obtain any non- roots of unity.

LEMMA 1.5 If divides , then is contained in

PROOF. Since we have so the result is clear

LEMMA 1.6 If and are relatively prime, then

and


37


(Recall the is the compositum of

PROOF. One checks easily that is a primitive root of unity, so that

Since this implies that

We know that has degree

over , so we must have

and

And thus that

PROPOSITION 1.2 For any and

And

here and denote the least common multiple and the greatest common divisor of and respectively.

PROOF. Write where the

are distinct primes. (We allow to be zero)

An entirely similar computation shows that

Mutual information measures the information transferred when

is sent and is received, and is defined as

In a noise-free channel, each is uniquely connected to the

corresponding , and so they constitute an input –output pair

for which

bits; that is, the transferred information is equal to the self-information that

corresponds to the input In a very noisy channel, the output

and input would be completely uncorrelated, and so

and also that is, there is no transference of information. In general, a given channel will operate between these two extremes. The mutual information is defined between the input and the output of a given channel. An average of the calculation of the mutual information for all input-output pairs of a given channel is the average mutual information:

bits per symbol . This calculation is done over the input and output alphabets. The average mutual information. The following


38


expressions are useful for modifying the mutual information expression:

Then

Where is usually called the equivocation. In a sense, the equivocation can be seen as the information lost in the noisy channel, and is a function of the backward conditional probability. The observation of an output

symbol provides bits of information. This difference is the mutual information of the channel. Mutual Information: Properties Since

The mutual information fits the condition

And by interchanging input and output it is also true that

Where

This last entropy is usually called the noise entropy. Thus, the information transferred through the channel is the difference between the output entropy and the noise entropy. Alternatively, it can be said that the channel mutual information is the difference between the number of bits needed for determining a given input symbol before knowing the corresponding output symbol, and the number of bits needed for determining a given input symbol after knowing the corresponding output symbol

As the channel mutual information expression is a difference between two quantities, it seems that this parameter can adopt negative values. However, and is spite of the fact that for some

can be larger than , this is not possible for the average value calculated over all the outputs:

Then

Because this expression is of the form

The above expression can be applied due to the factor

which is the product of two probabilities, so that it

behaves as the quantity , which in this expression is a dummy

variable that fits the condition . It can be concluded that the average mutual information is a non-negative number. It can also be equal to zero, when the input and the output are independent of each other. A related entropy called the joint entropy is defined as

Theorem 1.5: Entropies of the binary erasure channel (BEC) The BEC is defined with an alphabet of two inputs and three outputs, with symbol probabilities.


39


and transition probabilities

Lemma 1.7. Given an arbitrary restricted time-discrete, amplitude-

continuous channel whose restrictions are determined by sets and whose density functions exhibit no dependence on the state ,

let be a fixed positive integer, and an arbitrary probability

density function on Euclidean n-space. for the density

and . For any real number a, let

Then for each positive integer , there is a code such that

Where

Proof: A sequence such that

Choose the decoding set to be . Having chosen

and , select such that

Set , If the process does not terminate in a

finite number of steps, then the sequences and decoding sets

form the desired code. Thus assume that the process terminates after steps. (Conceivably ). We will show by showing that

. We proceed as follows.

Let

AlgorithmsLet A be a ring. Recall that an ideal a in A is a subset such that a is subgroup of A regarded as a group under addition;

The ideal generated by a subset S of A is the intersection of all ideals A containing a ----- it is easy to verify that this is in fact an

ideal, and that it consist of all finite sums of the form with

. When , we shall write

for the ideal it generates.

Let a and b be ideals in A. The set is an ideal, denoted by . The ideal generated by

is denoted by . Note that .

Clearly consists of all finite sums with and

, and if and , then

.Let be an ideal of A. The set of cosets of in A forms a ring , and is a

homomorphism . The map is a one to one correspondence between the ideals of and the ideals

of containing An ideal if prime if and

or . Thus is prime if and only if

is nonzero and has the property that

i.e., is an integral domain.

An ideal is maximal if and there does not exist an ideal contained strictly between and . Thus is maximal if and

only if has no proper nonzero ideals, and so is a field. Note that maximal prime. The ideals of are all of the form , with and ideals in and . To see this, note


40


that if is an ideal in and , then

and . This shows that with

and

Let be a ring. An -algebra is a ring together with a

homomorphism . A homomorphism of -algebra

is a homomorphism of rings such that

for all a A . An -algebra is said to be finitely generated ( or of finite-type over A) if there exist elements

such that every element of can be expressed as a

polynomial in the with coefficients in , i.e., such that the

homomorphism sending to is

surjective. A ring homomorphism is finite, and is

finitely generated as an A-module. Let be a field, and let be a

-algebra. If in , then the map is injective, we

can identify with its image, i.e., we can regard as a subring of

. If 1=0 in a ring R, the R is the zero ring, i.e., .

Polynomial rings. Let be a field. A monomial in is

an expression of the form . The total

degree of the monomial is . We sometimes abbreviate it by

. The elements of the polynomial ring

are finite sums

With the obvious notions of equality, addition and multiplication.

Thus the monomials from basis for as a -vector

space. The ring is an integral domain, and the only units in it are the nonzero constant polynomials. A polynomial

is irreducible if it is nonconstant and has only the

obvious factorizations, i.e., or is constant.

Division in . The division algorithm allows us to divide a

nonzero polynomial into another: let and be polynomials in

with then there exist unique polynomials

such that with either or deg < deg . Moreover, there is an algorithm for deciding whether

, namely, find and check whether it is zero. Moreover, the Euclidean algorithm allows to pass from finite set of generators

for an ideal in to a single generator by successively replacing each pair of generators with their greatest common divisor.

(Pure) lexicographic ordering (lex). Here monomials are ordered by lexicographic(dictionary) order. More precisely, let

and be two elements of ; then

and (lexicographic ordering) if, in the vector

difference , the left most nonzero entry is positive. For example,

. Note that this isn’t quite how the dictionary would order them: it would put

after . Graded reverse lexicographic order (grevlex). Here monomials are ordered by total degree, with

ties broken by reverse lexicographic ordering. Thus, if

, or and in the right most nonzero entry is negative. For example:

(total degree greater)

.

Orderings on . Fix an ordering on the monomials in

. Then we can write an element of

in a canonical fashion, by re-ordering its elements in decreasing order. For example, we would write

as

or

Let , in decreasing order:

Then we define.

The multidegree of to be multdeg( )= ;


41


The leading coefficient of to be LC( )= ;

The leading monomial of to be LM( ) = ;

The leading term of to be LT( ) =

For the polynomial the multidegree is (1,2,1),

the leading coefficient is 4, the leading monomial is , and

the leading term is . The division algorithm in

. Fix a monomial ordering in . Suppose given a

polynomial and an ordered set of polynomials; the

division algorithm then constructs polynomials and

such that Where either or no

monomial in is divisible by any of Step

1: If , divide into to get

If , repeat the process until

(different ) with not divisible by

. Now divide into , and so on, until

With not divisible by any

Step 2: Rewrite , and

repeat Step 1 with for :

(different ) Monomial ideals. In general, an ideal will contain a polynomial without containing the individual terms of the polynomial; for

example, the ideal contains but not

or .

DEFINITION 1.5. An ideal is monomial if

all with . PROPOSITION 1.3. Let be a monomial ideal, and let

. Then satisfies the condition

And is the -

subspace of generated by the .

Conversely, of is a subset of satisfying , then the k-

subspace of generated by is a monomial ideal.

PROOF. It is clear from its definition that a monomial ideal is

the -subspace of

generated by the set of monomials it contains. If and

. If a permutation is chosen uniformly and at random from the

possible permutations in then the counts of cycles of

length are dependent random variables. The joint distribution of

follows from Cauchy’s formula, and is given by

for .

Lemma1.7 For nonnegative integers

Proof. This can be established directly by exploiting cancellation

of the form when which occurs between the ingredients in Cauchy’s formula and the falling

factorials in the moments. Write . Then, with the

first sum indexed by and the last sum indexed

by via the correspondence we have

This last sum simplifies to the indicator corresponding

to the fact that if then for and a


42


random permutation in must have some cycle structure

. The moments of follow immediately as

We note for future reference that (1.4) can also be written in the form

Where the are independent Poisson-distribution random

variables that satisfy

The marginal distribution of cycle counts provides a formula for

the joint distribution of the cycle counts we find the

distribution of using a combinatorial approach combined with the inclusion-exclusion formula.

Lemma 1.8. For

Proof. Consider the set of all possible cycles of length

formed with elements chosen from so that

. For each consider the “property” of having that

is, is the set of permutations such that is one of

the cycles of We then have since the

elements of not in must be permuted among themselves. To use the inclusion-exclusion formula we need to

calculate the term which is the sum of the probabilities of the -fold intersection of properties, summing over all sets of

distinct properties. There are two cases to consider. If the properties are indexed by cycles having no elements in common,

then the intersection specifies how elements are moved by the

permutation, and there are permutations in the

intersection. There are such intersections. For the other case, some two distinct properties name some element in common, so no permutation can have both these properties, and the

-fold intersection is empty. Thus

Finally, the inclusion-exclusion series for the number of permutations having exactly properties is

Which simplifies to (1.1) Returning to the original hat-check problem, we substitute j=1 in (1.1) to obtain the distribution of the number of fixed points of a random permutation. For

and the moments of follow from (1.2) with In

particular, for the mean and variance of are both

equal to 1. The joint distribution of for any has an expression similar to (1.7); this too can be

derived by inclusion-exclusion. For any with

The joint moments of the first counts can be obtained directly from (1.2) and (1.3) by setting

The limit distribution of cycle counts

It follows immediately from Lemma 1.2 that for each fixed as

So that converges in distribution to a random variable

having a Poisson distribution with mean we use the notation

where to describe this. Infact, the limit random variables are independent.

Theorem 1.6 The process of cycle counts converges in

distribution to a Poisson process of with intensity . That is, as


43


Where the are independent Poisson-distributed

random variables with Proof. To establish the converges in distribution one shows that

for each fixed as

Error ratesThe proof of Theorem says nothing about the rate of convergence. Elementary analysis can be used to estimate this rate when . Using properties of alternating series with decreasing terms, for

It follows that

Since

We see from (1.11) that the total variation distance between the

distribution of and the distribution of

Establish the asymptotics of under conditions

and where

and as for some We start with the expression

and

Where refers to the quantity derived from . It thus

follows that for a constant ,

depending on and the and computable explicitly from (1.1)

– (1.3), if Conditions and are satisfied and if

from some since, under these

circumstances, both and tend to zero as In particular, for polynomials and square free polynomials, the relative error in this asymptotic approximation is

of order if

For and with

Where under Conditions and

Since, by the Conditioning Relation,

It follows by direct calculation that


44


Suppressing the argument from now on, we thus obtain

The

first sum is at most the third is bound by

Hence we may take

Required order under Conditions and if

If not, can be replaced by in the above, which has the required order, without the restriction on

the implied by . Examining the Conditions

and it is perhaps surprising to find that

is required instead of just that is, that we should need

to hold for some . A first observation

is that a similar problem arises with the rate of decay of as well.

For this reason, is replaced by . This makes it possible to

replace condition by the weaker pair of conditions and

in the eventual assumptions needed for to be of

order the decay rate requirement of order is

shifted from itself to its first difference. This is needed to obtain the right approximation error for the random mappings example. However, since all the classical applications make far

more stringent assumptions about the than are made in

. The critical point of the proof is seen where the initial

estimate of the difference . The

factor which should be small, contains a far tail

element from of the form which is only small

if being otherwise of order for any

since is in any case assumed. For this gives rise

to a contribution of order in the estimate of the

difference which, in the remainder of the proof, is translated into a contribution of order

for differences of the form

finally leading to a contribution of

order for any in Some improvement would seem to be possible, defining the function by

differences that are of the form

can be directly estimated, at a cost

of only a single contribution of the form Then, iterating the cycle, in which one estimate of a difference in point probabilities is improved to an estimate of smaller order, a bound of the form

for any could perhaps be attained, leading to a final error estimate

in order for any , to replace


45


This would be of the ideal order for large

enough but would still be coarser for small

With and as in the previous section, we wish to show that

Where for any

under Conditions and with . The proof uses sharper estimates. As before, we begin with the formula

Now we observe that

We have

The approximation in (1.2) is further simplified by noting that

and then by observing that


46


Combining the contributions of (1.2) –(1.3), we thus find tha

The quantity is seen to be of the order claimed under

Conditions and , provided that this

supplementary condition can be removed if is replaced

by in the definition of , has the required

order without the restriction on the implied by assuming that

Finally, a direct calculation now shows that

Example 1.0. Consider the point . For an arbitrary vector , the coordinates of the point are

equal to the respective coordinates of the vector

and . The vector r such as in the example is called the position vector or the radius vector of the point . (Or, in greater detail: is the radius-vector of w.r.t an origin O). Points are frequently specified by their radius-vectors. This presupposes the choice of O as the “standard origin”. Let us summarize. We

have considered and interpreted its elements in two ways: as points and as vectors. Hence we may say that we leading with the

two copies of = {points}, = {vectors} Operations with vectors: multiplication by a number, addition. Operations with points and vectors: adding a vector to a point

(giving a point), subtracting two points (giving a vector). treated in this way is called an n-dimensional affine space. (An “abstract” affine space is a pair of sets , the set of points and the set of vectors so that the operations as above are defined axiomatically). Notice that vectors in an affine space are also known as “free vectors”. Intuitively, they are not fixed at points

and “float freely” in space. From considered as an affine space

we can precede in two opposite directions: as an Euclidean

space as an affine space as a manifold.Going to the left means introducing some extra structure which will make the geometry richer. Going to the right means forgetting about part of the affine structure; going further in this direction will lead us to the so-called “smooth (or differentiable) manifolds”. The theory of differential forms does not require any extra geometry. So our natural direction is to the right. The Euclidean structure, however, is useful for examples and applications. So let us say a few words about it:

Remark 1.0. Euclidean geometry. In considered as an affine space we can already do a good deal of geometry. For example, we can consider lines and planes, and quadric surfaces like an ellipsoid. However, we cannot discuss such things as “lengths”, “angles” or “areas” and “volumes”. To be able to do so,

we have to introduce some more definitions, making a Euclidean space. Namely, we define the length of a vector

to be

After that we can also define distances between points as follows:

One can check that the distance so defined possesses natural properties that we expect: is it always non-negative and equals zero only for coinciding points; the distance from A to B is the same as that from B to A (symmetry); also, for three points, A, B and C, we

have (the “triangle inequality”). To define angles, we first introduce the scalar product of two vectors

Thus . The scalar product is also denote by dot:

, and hence is often referred to as the “dot product” . Now, for nonzero vectors, we define the angle between them by the equality

The angle itself is defined up to an integral multiple of

. For this definition to be consistent we have to ensure that the r.h.s. of (4) does not exceed 1 by the absolute value. This follows from the inequality

known as the Cauchy–Bunyakovsky–Schwarz inequality (various combinations of these three names are applied in different books). One of the ways of proving (5) is to consider the scalar square of


47


the linear combination where . As

is a quadratic polynomial in which is never negative, its discriminant must be less or equal zero. Writing this explicitly yields (5). The triangle inequality for distances also follows from the inequality (5).

Example 1.1. Consider the function (the i-th

coordinate). The linear function (the differential of )

applied to an arbitrary vector is simply .From these examples

follows that we can rewrite as

which is the standard form. Once again: the partial derivatives in

(1) are just the coefficients (depending on ); are linear functions giving on an arbitrary vector its coordinates

respectively. Hence

Theorem 1.7. Suppose we have a parametrized curve

passing through at and with the

velocity vector Then

Proof. Indeed, consider a small increment of the parameter

, Where . On the other hand, we have

for an arbitrary

vector , where when . Combining it together,

for the increment of we obtain

For a certain such that when (we

used the linearity of ). By the definition, this means that

the derivative of at is exactly . The statement of the theorem can be expressed by a simple formula:

To calculate the value Of at a point on a given vector

one can take an arbitrary curve passing Through at with

as the velocity vector at and calculate the usual derivative of

at .

Theorem 1.8. For functions ,

Proof. Consider an arbitrary point and an arbitrary vector

stretching from it. Let a curve be such that and

.

Hence

at and

at Formulae (1) and (2) then immediately follow from the corresponding formulae for the usual derivative Now, almost without change the theory generalizes to functions taking values in

instead of . The only difference is that now the differential

of a map at a point will be a linear function

taking vectors in to vectors in (instead of ) . For an

arbitrary vector

+

Where when . We have

and


48


In this matrix notation we have to write vectors as vector-columns.

Theorem 1.9. For an arbitrary parametrized curve in ,

the differential of a map (where ) maps

the velocity vector to the velocity vector of the curve

in

Proof. By the definition of the velocity vector,

Where when . By the definition of the differential,

Where when . we obtain

For some when . This precisely means that

is the velocity vector of . As every vector attached to a point can be viewed as the velocity vector of some curve passing through this point, this theorem gives a clear geometric picture of as a linear map on vectors.

Theorem 1.10 Suppose we have two maps and

where (open

domains). Let . Then the differential of the composite map is the composition of the differentials of and

Proof. We can use the description of the differential .Consider a

curve in with the velocity vector . Basically, we need

to know to which vector in it is taken by . the curve

. By the same theorem, it equals the image under of the Anycast Flow vector to the curve

in . Applying the theorem once again, we see that

the velocity vector to the curve is the image under of

the vector . Hence for an

arbitrary vector .

Corollary 1.0. If we denote coordinates in by

and in by , and write

Then the chain rule can be expressed as follows:

Where are taken from (1). In other words, to get

we have to substitute into (2) the expression for from (3). This can also be expressed by the following matrix formula:

i.e., if and are expressed by matrices of partial

derivatives, then is expressed by the product of these matrices. This is often written as


49


Or

Where it is assumed that the dependence of on is

given by the map , the dependence of on is

given by the map and the dependence of on is given by the composition .

Definition 1.6. Consider an open domain . Consider also

another copy of , denoted for distinction , with the

standard coordinates . A system of coordinates in the

open domain is given by a map where

is an open domain of , such that the following three conditions are satisfied :

(1) is smooth;

(2) is invertible;

(3) is also smooth

The coordinates of a point in this system are the standard

coordinates of In other words,

Here the variables are the “new” coordinates of the point

Example 1.2. Consider a curve in specified in polar coordinates as

We can simply use the chain rule. The map can be considered as the composition of the maps

. Then, by the chain rule, we have

Here and are scalar coefficients depending on , whence the

partial derivatives are vectors depending on point

in . We can compare this with the formula in the “standard”

coordinates: . Consider the vectors . Explicitly we have

From where it follows that these vectors make a basis at all points except for the origin (where ). It is instructive to sketch a picture, drawing vectors corresponding to a point as starting from

that point. Notice that are, respectively, the

velocity vectors for the curves

and . We can conclude that for an arbitrary curve given in polar coordinates the velocity vector will

have components if as a basis we take

A characteristic feature of the basis is that it is not “constant” but depends on point. Vectors “stuck to points” when we consider curvilinear coordinates.

Proposition 1.3. The velocity vector has the same appearance in all coordinate systems.Proof. Follows directly from the chain rule and the

transformation law for the basis .In particular, the elements of

the basis (originally, a formal notation) can be understood directly as the velocity vectors of the coordinate lines


50


(all coordinates but are fixed). Since we now know how to handle velocities in arbitrary coordinates, the

best way to treat the differential of a map is by its action on the velocity vectors. By definition, we set

Now is a linear map that takes vectors attached to a point

to vectors attached to the point

In particular, for the differential of a function we always have

Where are arbitrary coordinates. The form of the differential does not change when we perform a change of coordinates.

Example 1.3 Consider a 1-form in given in the standard coordinates:

In the polar coordinates we will have

, hence

Substituting into , we get

Hence is the formula for in the polar coordinates. In particular, we see that this is again a 1-form, a linear combination of the differentials of coordinates with functions as coefficients. Secondly, in a more conceptual way, we can define a 1-form in a domain as a linear function on vectors at every

point of :

If , where . Recall that the differentials of functions were defined as linear functions on vectors (at every

point), and at every point .

Theorem 1.9. For arbitrary 1-form and path , the integral

does not change if we change parametrization of provide the orientation remains the same.

Proof: Consider and As

=

Let be a rational prime and let We write for

or this section. Recall that has degree over

We wish to show that Note that is a root of

and thus is an algebraic integer; since is a ring we

have that We give a proof without assuming unique factorization of ideals. We begin with some norm and trace

computations. Let be an integer. If is not divisible by then

is a primitive root of unity, and thus its conjugates are

Therefore

If does divide then so it has only the one conjugate

1, and By linearity of the trace, we find that

We also need to compute the norm of . For this, we use the factorization

Plugging in shows that


51


Since the are the conjugates of this shows that

The key result for determining the ring of

integers is the following.

LEMMA 1.9

Proof. We saw above that is a multiple of in so

the inclusion is immediate. Suppose now

that the inclusion is strict. Since is an ideal of

containing and is a maximal ideal of , we must have

Thus we can write

For some That is, is a unit in

COROLLARY 1.1 For any PROOF. We have

Where the are the complex embeddings of (which we are really viewing as automorphisms of ) with the usual ordering.

Furthermore, is a multiple of in for every

Thus

Since the trace is also a rational integer.

PROPOSITION 1.4 Let be a prime number and let

be the cyclotomic field. Then

Thus is an

integral basis for .

PROOF. Let and write

With Then

By the linearity of the trace and our above calculations we find that

We also have

so Next consider the algebraic integer

This is an algebraic

integer since is. The same argument as above shows

that and continuing in this way we find that all of the are in . This completes the proof.

Example 1.4 Let , then the local ring is simply the

subring of of rational numbers with denominator relatively

prime to . Note that this ring is not the ring of -

adic integers; to get one must complete . The usefulness

of comes from the fact that it has a particularly simple ideal

structure. Let be any proper ideal of and consider the ideal

of We claim that That is,

that is generated by the elements of in It is clear

from the definition of an ideal that To prove the other inclusion, let be any element of . Then we can write

where and In particular,

(since and is an ideal), so and so

Since this implies that

as claimed.We can use this fact to

determine all of the ideals of Let be any ideal of

and consider the ideal factorization of in write it as

For some and some ideal relatively prime

to we claim first that We now find that

Since

Thus every ideal of has the form for some it

follows immediately that is noetherian. It is also now clear

that is the unique non-zero prime ideal in .

Furthermore, the inclusion Since

this map is also surjection, since the residue


52


class of (with and ) is the image of

in which makes sense since is invertible in

Thus the map is an isomorphism. In particular, it is now

abundantly clear that every non-zero prime ideal of is

maximal. To show that is a Dedekind domain, it

remains to show that it is integrally closed in . So let be

a root of a polynomial with coefficients in write this

polynomial as With and

Set Multiplying by we find

that is the root of a monic polynomial with coefficients in

Thus since we have

. Thus is integrally close in

COROLLARY 1.2. Let be a number field of degree and

let be in then PROOF. We assume a bit more Galois theory than usual for this

proof. Assume first that is Galois. Let be an element of

It is clear that since

this shows that

. Taking the product over all

we have

Since is a

rational integer and is a free -module of rank

Will have order therefore

This completes the proof. In the general case, let be the Galois

closure of and set

A. Spatial AnalysisSpatial Analysis of people suffering from Cancer in North

America and the trend in Geo Location. Spatial Analysis is to measure properties and relationship with spatial localization and the events like Brain Cancer in America. The model processes define the distribution of spread of cancer in space.

Taxonomy used are Events, Point Patterns to express occurrences of Cancer patient as points in space listed as Point Processes and give the localization coordinates. This study developed the modelling process for exploratory analysis to provide graphs, maps and spatial patterns.

In Point Pattern Analysis the object of interest is the spatial location of cancer events as the type of cancer and the numbers associated with Mortality. Objective is to study the spatial distribution and develop testing hypothesis about the observed and forecast pattern.

The model uses the geostatistics techniques to define homogeneous bahavior on the spatial correlation data structure in geolocation.

Spatial Autocorrelation is the spatial dependency based on computation framework, this is to measure relationship between two random variables, but are applying the concept on multiple variable the distinguish Brain Tumor Types, Nervous Cancer Types, Location and Influence Factors. Verifying spatial dependency varies based on comparative analysis of population sample and nearest points.

Fig 1 : Delaunay Tetrahedra Volume

Fig. 2: F Function ( Cumulative Sample Point to Nearest Cell Distances)


53


Fig 3: G Function (Cumulative Nearest Neighbor Distribution)

Fig 4: K Function (Cumulative Density)

Fig 5 : Near Neighbors

Fig 6 : Three Dimension Autocorrelation and Histogram

Fig 7 : Voronio Domain

Fig 8 : Autocorrelation Histogram

Fig 9 : Voronio Domain Director Vector and Histogram

Fig 10 : North America Cancer patient distribution

Cancer Types

0-19 20+ Benign/Borderli

ne

Malignant

Benign/Borderli

ne

Malignant

Total 1.6(1.5-1.8)

3.4(3.3-3.6)

12.1(11.9-12.3)

10(9.8-10.2)

Tumors of Neuroepthelial

.6(0.5-0.6)

3.1(2.9-

.4(0.4-0.5)

8.5(8.3-


54


Tissue 3.3) 8.7)

Pilocytic astrocytoma

.8(0.7-0.9)

.1(0.1-0.2)

Diffuse astrocytoma

.1(0.0-0.1)

.2(0.1-0.2)

Anaplastic astrocytoma

.1(0.1-0.2)

.6(0.6-0.6)

Unique astrocytoma variants

.1(0.0-0.1)

.1(0.0-0.1)

.1(0.1-0.1)

0.(0.0-0.0)

Astrocytoma, NOS

.2(0.2-0.3)

.6(0.5-0.6)

Glioblastoma.2

(0.1-0.2)

5.3(5.2-5.5)

Oligodendroglioma

.1(0.0-0.1)

.4(0.3-0.4)

Anaplastic oligodendroglioma

.2(0.1-0.2)

Ependymoma/anaplastic ependymoma

.3(0.2-0.3)

.2(0.2-0.3)

Ependymoma variants

0.(0.0-0.1)

.1(0.1-0.1)

Mixed glioma.3

(0.3-0.3)

Glioma malignant, NOS

.5(0.5-0.6)

.4(0.4-0.5)

Choroid plexus .1(0.1-0.1) ~ 0.

(0.0-0.0) ~

Neuroepithelial0.

(0.0-0.0)

Neuronal/glial, neuronal

.4(0.3-0.4)

.1(0.0-0.1)

.2(0.1-0.2)

0.(0.0-0.0)

Pineal parenchymal

0.(0.0-0.1)

Embryonal/primitive/medulloblastoma

.7(0.6-0.8)

.1(0.1-0.1)

Tumors of Cranial and Spinal Nerves

.3(0.2-0.3)

2.1(2.1-2.2)

0.(0.0-0.0)

Nerve sheath, benign and malignant

.3(0.2-0.3)

2.1(2.1-2.2)

0.(0.0-0.0)

Tumors of the Meninges

.2(0.1-0.2)

5.3(5.1-5.4)

.2(0.2-0.2)

Meningioma .1(0.1-0.1)

5(4.8-5.1)

.2(0.1-0.2)

Other mesenchymal

.1(0.1-0.1)

0.(0.0-0.0)

Hemangioblastoma

0.(0.0-0.1)

.2(0.2-0.3)

Lymphomas and Hematopoietic Neoplasms

.7(0.6-0.7)

Germ Cell Tumors.2

(0.2-0.3)

0.(0.0-0.0)

.1(0.1-0.1)

Cancer Types

0-19 20+ Benign/Borderli

ne

Malignant

Benign/Borderli

ne

Malignant

Tumors of Sellar Region

.4(0.3-0.5)

3.5(3.4-3.6)

0.(0.0-0.0)

Pituitary .2(0.2-0.3)

3.3(3.2-3.4)

0.(0.0-0.0)

Craniopharyngioma

.2(0.1-0.2)

.2(0.1-0.2)

Local Extensions from Regional Tumors

0.(0.0-0.0)

Unclassified Tumors

.2(0.2-0.3)

.1(0.0-0.1)

.7(0.7-0.8)

.5(0.5-0.6)

Hemangioma .1(0.1-0.1)

.2(0.2-0.2)

Neoplasm, .1 .1 .5 .5


55


unspecified (0.1-0.2) (0.0-0.1) (0.5-0.6) (0.5-

0.6)All other

Fig11: Crude Incidence Rate [4]


56


Fig. 12


57


Classification of Brain Cancer

IV. CONCLUSIONS

Cancer patients in America is reducing and especially Brain Cancer percentage is in control and not increase as compared to Lung Cancer. Next work is to layout the framework for epidemic models

REFERENCES[1] Francis P Boscoe, Mary H Ward and Peggy Reynolds,Current practices in

spatial analysis of cancer data: data characteristics and data sources for geographic studies of cancer,International Journal of Health Geographics 2004, 3:28

[2] Burkitt DP: Geography of a disease: purpose and possibilities from geographical medicine. In Biocultural aspects of disease Edited by: Rothschild HR. New York, Academic Press; 1981.

[3] Gould P, Wallace R: Spatial structures and scientific paradoxes in the AIDS pandemic. Geografiska Annaler B 1994, 76:105-116.

[4] http://www.braintumor.org/news/press-kit/brain-tumor-facts.html Central Brain Tumor Registry of the United States statistics report


58

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